Computing Exact Geometric Predicates Using Modular Arithmetic with Single Precision

We propose an efficient method that determines the sign of a multivariate polynomial expression with integer coefficients. This is a central operation on which the robustness of many geometric algorithms depends. Our method relies on modular computations, for which comparisons are usually thought to require multiprecision. Our novel technique of recursive relaxation of the moduli enables us to carry out sign determination and comparisons by using only floating point computations in single precision. This leads us to propose a hybrid symbolic-numeric approach to exact arithmetic. The method is highly parallelizable and is the fastest of all known multiprecision methods from a complexity point of view. As an application, we show how to compute a few geometric predicates that reduce to matrix determinants and we discuss implementation efficiency, which can be enhanced by arithmetic filters. We substantiate these claims by experimental results and comparisons to other existing approaches. Our..

Computing exact geometric predicates using modular arithmetic with single precision

Theme 2 - Genie logiciel et calcul symbolique - Projet PrismeSIGLEAvailable from INIST (FR), Document Supply Service, under shelf-number : 14802 E, issue : a.1997 n.3213 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

Degree-Driven Design of Geometric Algorithms for Point Location, Proximity, and Volume Calculation

Correct implementation of published geometric algorithms is surprisingly difficult. Geometric algorithms are often designed for Real-RAM, a computational model that provides arbitrary precision arithmetic operations at unit cost. Actual commodity hardware provides only finite precision and may result in arithmetic errors. While the errors may seem small, if ignored, they may cause incorrect branching, which may cause an implementation to reach an undefined state, produce erroneous output, or crash. In 1999 Liotta, Preparata and Tamassia proposed that in addition to considering the resources of time and space, an algorithm designer should also consider the arithmetic precision necessary to guarantee a correct implementation. They called this design technique degree-driven algorithm design. Designers who consider the time, space, and precision for a problem up-front arrive at new solutions, gain further insight, and find simpler representations. In this thesis, I show that degree-driven design supports the development of new and robust geometric algorithms. I demonstrate this claim via several new algorithms. For n point sites on a UxU grid I consider three problems. First, I show how to compute the nearest neighbor transform in O(U^2) expected time, O(U^2) space, and double precision. Second, I show how to create a data structure in O(n log Un) expected time, O(n) expected space, and triple precision that supports O(log n) time and double precision post-office queries. Third, I show how to compute the Gabriel graph in O(n^2) time, O(n^2) space and double precision. For computing volumes of CSG models, I describe a framework that uses a minimal set of predicates that use at most five-fold precision. The framework is over 500x faster and two orders of magnitude more accurate than a Monte Carlo volume calculation algorithm.Doctor of Philosoph

Geometric algorithms for algebraic curves and surfaces

This work presents novel geometric algorithms dealing with algebraic curves and surfaces of arbitrary degree. These algorithms are exact and complete — they return the mathematically true result for all input instances. Efficiency is achieved by cutting back expensive symbolic computation and favoring combinatorial and adaptive numerical methods instead, without spoiling exactness in the overall result. We present an algorithm for computing planar arrangements induced by real algebraic curves. We show its efficiency both in theory by a complexity analysis, as well as in practice by experimental comparison with related methods. For the latter, our solution has been implemented in the context of the Cgal library. The results show that it constitutes the best current exact implementation available for arrangements as well as for the related problem of computing the topology of one algebraic curve. The algorithm is also applied to related problems, such as arrangements of rotated curves, and arrangments embedded on a parameterized surface. In R3, we propose a new method to compute an isotopic triangulation of an algebraic surface. This triangulation is based on a stratification of the surface, which reveals topological and geometric information. Our implementation is the first for this problem that makes consequent use of numerical methods, and still yields the exact topology of the surface.Diese Arbeit stellt neue Algorithmen für algebraische Kurven und Flächen von beliebigem Grad vor. Diese Algorithmen liefern für alle Eingaben das mathematisch korrekte Ergebnis. Wir erreichen Effizienz, indem wir aufwendige symbolische Berechnungen weitesgehend vermeiden, und stattdessen kombinatorische und adaptive numerische Methoden einsetzen, ohne die Exaktheit des Resultats zu zerstören. Der Hauptbeitrag ist ein Algorithmus zur Berechnung von planaren Arrangements, die durch reelle algebraische Kurven induziert sind. Wir weisen die Effizienz des Verfahrens sowohl theoretisch durch eine Komplexitätsanalyse, als auch praktisch durch experimentelle Vergleiche nach. Dazu haben wir unser Verfahren im Rahmen der Softwarebibliothek Cgal implementiert. Die Resultate belegen, dass wir die zur Zeit beste verfügbare exakte Software bereitstellen. Der Algorithmus wird zur Arrangementberechnung rotierter Kurven, oder für Arrangements auf parametrisierten Oberflächen eingesetzt. Im R3 geben wir ein neues Verfahren zur Berechnung einer isotopen Triangulierung einer algebraischen Oberfläche an. Diese Triangulierung basiert auf einer Stratifizierung der Oberfläche, die topologische und geometrische Informationen berechnet. Unsere Implementierung ist die erste für dieses Problem, welche numerische Methoden konsequent einsetzt, und dennoch die exakte Topologie der Oberfläche liefert

Exact Algorithms For Circles On The Sphere

We describe exact representations and algorithms for geometric operations on general circles and circular arcs on the sphere, using integer homogeneous coordinates. The algorithms include testing a point against a circle, computing the intersection of two circles, and ordering three arcs out of the same point. These tools support robust and efficient operations on maps overs the sphere, such as point location and map overlay, and provide a reliable framework for robotics, geographic information systems, and other geometric applications.113267290Aho, A., Johnson, D.S., Karp, R.M., Kosaraju, S.R., McGeoch, C.C., Papadimitriou, C.H., Pevzner, P., (1996) "Theory of Computing: Goals and Directions,", , ManuscriptAndrade, M.V.A., (1999) Representação e Manipulação Exatas de Mapas Esféricos, , Ph. D. Thesis, Institute of Computing, University of Campinas, (In Portuguese)Brönnimann, H., Emiris, I., Pan, V., Pion, S., Computing exact geometric predicates using modular arithmetic with single precision (1997) Proc. 13th Ann. ACM Sympos. on Comput. Geom., pp. 174-182Brönnimann, H., Yvinec, M., Efficient exact evaluation of signs of determinants (1997) Proc. 13th Ann. ACM Sympos. on Comput. Geom., pp. 166-173(1996) Applications Challenges to Computational Geometry, , Technical Report TR-521-96, Princeton UniversityCox, D., Little, J., O'Shea, D., (1992) Ideals, Varieties, and Algorithms, , Springer-VerlagEdelsbrunner, H., Guibas, L., Topologically Sweeping an Arrangement (1989) J. Comput. System Sci., 38, pp. 165-194Finke, U., Hinrichs, K., Overlaying Simply Connected Planar Subdivisions in Linear Time (1995) Proc. 11th Ann. ACM Symp. on Comput. Geom., pp. 119-126Fortune, S., Van Wyk, C.J., Efficient exact arithmetic for computational geometry (1993) Proc. 9th Ann. ACM Sympos. Comput. Geom., pp. 163-172Goodrich, M.T., Guibas, L.J., Hershberger, J., Tannenbaum, P.J., Snap Rounding Line Segments Efficiently in Two and Three Dimensions (1997) Proc. 13th Ann. ACM Symp. on Comput. Geom., pp. 284-293Granlund, T., The GNU Multiple Precision Arithmetic Library, , http://www.gnu.org/manual/gmp/gmp.html, Free Software FoundationGreene, D.H., Yao, F.F., Finite-resolution computational geometry (1986) Proc. 27th Ann. IEEE Sympos. Found. Comput. Sci., pp. 143-152Guibas, L., Marimont, D., Rounding arrangements dynamically (1995) Proc. 11th Ann. ACM Sympos. Comput. Geom., pp. 190-199Guibas, L., Stolfi, J., Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi Diagrams (1985) ACM Transactions on Graphics, 4 (2), pp. 74-123Halperin, D., Shelton, C.R., A perturbation scheme for spherical arrangements with application to molecular modeling (1998) Computational Geometry: Theory and Applications, 10 (4), pp. 273-288Hoffmann, C.M., The problems of accuracy and robustness in geometric computation (1989) IEEE Computer, 22 (3), pp. 31-42Maguire, D.J., Goodchild, M.F., Rhind, D., (1991) Geographical Information Systems - Principles and Applications, , John Wiley & SonsNelson, G., (1991) Systems Programming with Modula-3, , Prentice HallStolfi, J., (1991) Oriented Protective Geometry - A Framework for Geometric Computations, , Academic PressWu, P.Y.F., Franklin, W.R., A Logic Programming Approach to Cartographic Map Overlay (1990) Canadian Computational Intelligence Journal, 6 (2), pp. 61-70Yap, C.K., Towards Exact Geometric Computation (1993) Proc. 5th Canad. Conf. Comput. Geom, pp. 405-419Yap, C.K., Dubé, T., The exact computation paradigm (1995) Computing in Euclidean Geometry, Volume 1 of Lecture Notes Series on Computing, pp. 452-492. , D.-Z Du and F. K. Hwang, editors, World Scientific Press, Singapore, 2nd. editio